Volume of a Prism - Formula, Derivation, Definition, Examples
A prism is an important shape in geometry. The figure’s name is derived from the fact that it is created by taking a polygonal base and expanding its sides till it intersects the opposing base.
This blog post will take you through what a prism is, its definition, different types, and the formulas for volume and surface area. We will also take you through some examples of how to use the information given.
What Is a Prism?
A prism is a three-dimensional geometric figure with two congruent and parallel faces, called bases, which take the shape of a plane figure. The other faces are rectangles, and their number relies on how many sides the identical base has. For example, if the bases are triangular, the prism would have three sides. If the bases are pentagons, there will be five sides.
Definition
The properties of a prism are interesting. The base and top both have an edge in common with the other two sides, making them congruent to one another as well! This means that all three dimensions - length and width in front and depth to the back - can be broken down into these four entities:
A lateral face (implying both height AND depth)
Two parallel planes which make up each base
An illusory line standing upright across any given point on any side of this shape's core/midline—usually known collectively as an axis of symmetry
Two vertices (the plural of vertex) where any three planes join
Types of Prisms
There are three primary kinds of prisms:
Rectangular prism
Triangular prism
Pentagonal prism
The rectangular prism is a common type of prism. It has six sides that are all rectangles. It looks like a box.
The triangular prism has two triangular bases and three rectangular sides.
The pentagonal prism consists of two pentagonal bases and five rectangular sides. It looks a lot like a triangular prism, but the pentagonal shape of the base stands out.
The Formula for the Volume of a Prism
Volume is a measure of the sum of space that an thing occupies. As an crucial shape in geometry, the volume of a prism is very relevant in your studies.
The formula for the volume of a rectangular prism is V=B*h, assuming,
V = Volume
B = Base area
h= Height
Consequently, since bases can have all sorts of figures, you will need to retain few formulas to determine the surface area of the base. Despite that, we will touch upon that afterwards.
The Derivation of the Formula
To derive the formula for the volume of a rectangular prism, we have to look at a cube. A cube is a three-dimensional item with six faces that are all squares. The formula for the volume of a cube is V=s^3, where,
V = Volume
s = Side length
Right away, we will take a slice out of our cube that is h units thick. This slice will create a rectangular prism. The volume of this rectangular prism is B*h. The B in the formula refers to the base area of the rectangle. The h in the formula stands for height, that is how dense our slice was.
Now that we have a formula for the volume of a rectangular prism, we can use it on any kind of prism.
Examples of How to Use the Formula
Since we understand the formulas for the volume of a rectangular prism, triangular prism, and pentagonal prism, let’s put them to use.
First, let’s calculate the volume of a rectangular prism with a base area of 36 square inches and a height of 12 inches.
V=B*h
V=36*12
V=432 square inches
Now, let’s try another question, let’s work on the volume of a triangular prism with a base area of 30 square inches and a height of 15 inches.
V=Bh
V=30*15
V=450 cubic inches
As long as you have the surface area and height, you will calculate the volume without any issue.
The Surface Area of a Prism
Now, let’s talk regarding the surface area. The surface area of an object is the measure of the total area that the object’s surface comprises of. It is an crucial part of the formula; thus, we must understand how to find it.
There are a several distinctive methods to find the surface area of a prism. To figure out the surface area of a rectangular prism, you can use this: A=2(lb + bh + lh), assuming,
l = Length of the rectangular prism
b = Breadth of the rectangular prism
h = Height of the rectangular prism
To figure out the surface area of a triangular prism, we will utilize this formula:
SA=(S1+S2+S3)L+bh
where,
b = The bottom edge of the base triangle,
h = height of said triangle,
l = length of the prism
S1, S2, and S3 = The three sides of the base triangle
bh = the total area of the two triangles, or [2 × (1/2 × bh)] = bh
We can also utilize SA = (Perimeter of the base × Length of the prism) + (2 × Base area)
Example for Finding the Surface Area of a Rectangular Prism
First, we will work on the total surface area of a rectangular prism with the following data.
l=8 in
b=5 in
h=7 in
To figure out this, we will put these values into the corresponding formula as follows:
SA = 2(lb + bh + lh)
SA = 2(8*5 + 5*7 + 8*7)
SA = 2(40 + 35 + 56)
SA = 2 × 131
SA = 262 square inches
Example for Computing the Surface Area of a Triangular Prism
To find the surface area of a triangular prism, we will find the total surface area by following similar steps as before.
This prism consists of a base area of 60 square inches, a base perimeter of 40 inches, and a length of 7 inches. Therefore,
SA=(Perimeter of the base × Length of the prism) + (2 × Base Area)
Or,
SA = (40*7) + (2*60)
SA = 400 square inches
With this information, you should be able to work out any prism’s volume and surface area. Check out for yourself and see how easy it is!
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